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Legendre’s theorem on angles of triangle

Adrien-Marie Legendre has proved some theorems concerning the sum of the angles of triangle. Here we give one of them, being the inverse of the theorem in the entry “sum of angles of triangle in Euclidean geometry”.

Theorem. If the sum of the interior angles of every triangle equals straight angle, then the parallel postulate is true, i.e., in the plane determined by a line and a point outwards it there is exactly one line through the point which does not intersect the line.

Proof. We consider a line aaa and a point BBB not belonging to aaa. Let B⁢ABABA be the normal line of aaa
(with A∈aAaA\in a) and bbb be the normal line of B⁢ABABA through the point BBB. By the supposition of the theorem, bbb does not intersect aaa.

We will show that in the plane determined by the line aaa and the point BBB, there are through BBB no other lines than bbb not intersecting the line aaa. For this purpose, we choose through BBB a line b′superscriptbnormal-′b^{{\prime}} which differs from bbb; let the line b′superscriptbnormal-′b^{{\prime}} form with B⁢ABABA an acute angle ββ\beta.

We determine on the line aaa a point A1subscriptA1A_{1} such that A⁢A1=A⁢BAsubscriptA1ABAA_{1}=AB. By the supposition of the theorem, in the isosceles right triangle B⁢A⁢A1BAsubscriptA1BAA_{1} we have

Next we determine on aaa a second point A2subscriptA2A_{2} such that
A1⁢A2=A1⁢BsubscriptA1subscriptA2subscriptA1BA_{1}A_{2}=A_{1}B. By the supposition of the theorem, in the isosceles triangle B⁢A1⁢A2BsubscriptA1subscriptA2BA_{1}A_{2} we have

They form a geometric sequence with the common ratio
r=12r12r=\frac{1}{2}. When nnn is sufficiently great, the member αnsubscriptαn\alpha_{n} is less than any given positive angle. As we have so much triangles B⁢An-1⁢AnBsubscriptAn1subscriptAnBA_{{n-1}}A_{n} that
αn<π2-βsubscriptαnπ2β\alpha_{n}<\frac{\pi}{2}\!-\!\beta, then

Then the line b′superscriptbnormal-′b^{{\prime}} falls after penetrating BBB into the inner territory of the triangle A⁢B⁢AnABsubscriptAnABA_{n}. Thereafter it must leave from there and thus intersect the side A⁢AnAsubscriptAnAA_{n} of this triangle. Accordingly, b′superscriptbnormal-′b^{{\prime}} intersects the line aaa.

The above reasoning is possible for each line b′≠bsuperscriptbnormal-′bb^{{\prime}}\neq b through BBB. Consequently, the parallel axiom is in force.